Light propagation phenomena such as vacuum birefringence are encoded in contributions linear in the incident probe field; vacuum-fluctuation-mediated corrections to probe photon propagation manifest themselves in signal photons supplementing the outgoing probe field with quantum corrections.
Upon limitation to the leading effective interaction in Eq. (REF ), the current (REF ) is cubic in the driving field. This immediately implies the vacuum birefringence phenomenon studied here to be linear in the plane wave probe and quadratic in the background magnetic field.
Besides, the signal photon current in position space \(j_\nu (x)\)  is manifestly real-valued.
Being interested in the signal to be detected far outside the interaction region at vanishing magnetic field \(B\) , the electric field of the induced signal can then be expressed as [1]}
\(\vec{e}_{\rm s}(x)={\rm Re}\int \frac{{\rm d}^3k}{(2\pi )^3}\,{\rm e}^{{\rm i}kx}\,\underbrace{\bigl (\vec{j}(k)-\hat{\vec{k}}j^0(k)\bigr )}_{=:\vec{e}_{\rm s}(k)}\Big |_{k^0=|\vec{k}|}\,, \) 
